Geometry of Grief: Reflections on Mathematics, Loss, and Life

By Michael Frame

In this profound and hopeful book, a mathematician and celebrated teacher shows how mathematics may help all of us—even the math-averse—to understand and cope with grief.
 
We all know the euphoria of intellectual epiphany—the thrill of sudden understanding. But coupled with that excitement is a sense of loss: a moment of epiphany can never be repeated. In Geometry of Grief, mathematician Michael Frame draws on a career’s worth of insight—including his work with a pioneer of fractal geometry Benoit Mandelbrot—and a gift for rendering the complex accessible as he delves into this twinning of understanding and loss. Grief, Frame reveals, can be a moment of possibility.

Frame investigates grief as a response to an irrevocable change in circumstance. This reframing allows us to see parallels between the loss of a loved one or a career and the loss of the elation of first understanding a tricky concept. From this foundation, Frame builds a geometric model of mental states. An object that is fractal, for example, has symmetry of magnification: magnify a picture of a mountain or a fern leaf—both fractal—and we see echoes of the original shape. Similarly, nested inside great loss are smaller losses. By manipulating this geometry, Frame shows us, we may be able to redirect our thinking in ways that help reduce our pain. Small‐scale losses, in essence, provide laboratories to learn how to meet large-scale losses.

Interweaving original illustrations, clear introductions to advanced topics in geometry, and wisdom gleaned from his own experience with illness and others’ remarkable responses to devastating loss, Frame’s poetic book is a journey through the beautiful complexities of mathematics and life. With both human sympathy and geometrical elegance, it helps us to see how a geometry of grief can open a pathway for bold action.

• Language: English
• Publisher: University of Chicago Press
• Release date: Sep 8, 2021
• ISBN: 9780226801087

Book preview

Cover Page for Geometry of Grief

Geometry of Grief

Geometry of Grief

Reflections on Mathematics, Loss, and Life

Michael Frame

The University of Chicago Press

Chicago and London

The University of Chicago Press, Chicago 60637

The University of Chicago Press, Ltd., London

© 2021 by Michael Frame

All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission, except in the case of brief quotations in critical articles and reviews. For more information, contact the University of Chicago Press, 1427 E. 60th St., Chicago, IL 60637.

Published 2021

Printed in the United States of America

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ISBN-13: 978-0-226-80092-9 (cloth)

ISBN-13: 978-0-226-80108-7 (e-book)

DOI: https://doi.org/10.7208/chicago/9780226801087.001.0001

Library of Congress Cataloging-in-Publication Data

Names: Frame, Michael, author.

Title: Geometry of grief : reflections on mathematics, loss, and life / Michael Frame.

Description: Chicago : University of Chicago Press, 2021. | Includes bibliographical references and index.

Identifiers: LCCN 2021007566 | ISBN 9780226800929 (cloth) | ISBN 9780226801087 (ebook)

Subjects: LCSH: Fractals. | Grief. | Geometry. | Mathematics—Social aspects.

Classification: LCC QA614.86 .F796 2021 | DDC 514/.742—dc23

LC record available at https://lccn.loc.gov/2021007566

This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).

Contents

Prologue

1: Geometry

2: Grief

3: Beauty

4: Story

5: Fractal

6: Beyond

Appendix: More Math

Acknowledgments

Notes

Index

Prologue

Dad, that’s really scary.

Look for the very brightest one in the sky.

Beside the tree, about halfway up? Is that it, Ruthie?

That’s it. That’s Venus. It’s a planet, a whole world, almost as big as the earth. And it’s cloudy all the time. No one has ever seen land on Venus.

If it’s cloudy all the time, Venus must be cold.

Not necessarily. Venus is closer to the sun than Earth is. Maybe the clouds hold in the heat and it’s very hot there.

Oh, I see. The sky is clear tonight, so we’ll get cooler than we would if it was cloudy.

That’s right, Mikey. Do you want to go inside now?

Are there other planets in the sky?

Not tonight.

Can we stay outside and watch lightning bugs?

Sure.

This was an evening late in the summer of 1958. The sky, purple deepening to indigo, showed a few pinpoint stars and a much brighter dot, Venus. We’d had dinner with my grandmother and my Aunt Ruthie, my dad’s sister, at their house in South Charleston, West Virginia. I was seven, my sister Linda was four, our brother Steve was two. Only Ruthie and I were in the backyard. The others were on the front porch, visiting, Mom called it. We lived in St. Albans, West Virginia, only about eight miles away, and saw my grandmother and Ruthie often. Just why the adults visited wasn’t clear to me. What could they talk about? They just gossiped about their neighbors and other family members.

Ruthie and I were different. That afternoon we’d sat in the kitchen garden, and the purposeful march of ants and the random jumps of grasshoppers entranced us. I constructed elaborate natural histories to explain their behaviors; Ruthie proposed much simpler alternatives. She never used the term Occam’s razor, but she had begun to teach me the beauty of simple explanations. And also the likelihood of economy: a Rube Goldberg machine—a complicated contraption that takes up a whole room and performs a simple task like cracking an egg—has many points of potential failure. My complex pathways were good mental exercises, maybe, but did I really think nature would be that silly? Years later, I understood that Ruthie had started me on the path to becoming a scientist. She thought curiosity is the most important trait of the mind; that the curiosity of a child, the twists and turns of young logic when the child unpacks aspect and dynamic of the wide world, is the most beautiful thing an adult can see. Mom and Dad, grandparents, other aunts and uncles, encouraged curiosity, but Ruthie cultivated it, mixed in some skepticism, and always found a book for me to read about the topic of current interest. Ruthie set me on the way that, sixty years later, has led me to write this story.

In elementary school career discussions, against my classmates’ police officer, firefighter, and park ranger (astronaut wasn’t a career then—yes, I’m ancient), I offered physicist or mathematician or astronomer. But really, at that age every kid is a naturalist. A summer morning in neighboring woods revealed wonders without end. The optimism of childhood knew no bounds. My parents’ finances, though limited, afforded opportunity for creative explorations. To measure the output of a thermocouple (a copper wire and a steel wire twisted together that convert heat into a weak electrical current), the father of another student bought an expensive multimeter. I made a galvanometer: two magnetized needles stuck through a small cardboard rectangle suspended by thread in a coil of wire. Who had more fun detecting the tiny current?

Ruthie didn’t help me design the experiments—Dad did that, and let me set up a small lab in a corner of his workshop—but Ruthie helped me realize that I could do experiments and answer some of my own questions.

Late in my eleventh year, Ruthie got sick. Hodgkin’s lymphoma, survivable now but not so much in the early 1960s. She was treated, with the chemotherapy drug Mustargen, I believe, but lived only a few more months in some misery and died early in my twelfth year. I visited Ruthie when she was sick, but I couldn’t do much. I stood beside her bed, rested my little hand on her forearm and tried to talk with her. But I couldn’t think of anything to say. At home after these visits, Mom hugged me, stroked my hair. I knew I should have talked more with Ruthie. She had done so much for me, and she needed me now. She needed me to talk with her because I was her favorite. Later I understood that Mom was working through her own grief. She knew the situation far better than I did, knew this disease would win and Ruthie would lose. Dad began to talk with me about his sister’s illness. He was straightforward: Ruthie was going to die. I appreciated his honesty. No nonsense about Ruthie going away, or—worse—going to live with the angels.¹ Her life would end, and soon. This isn’t fair. There’s so much more for Ruthie and me to do. She promised we’d get a telescope to look at the planets. I’ve saved my allowance for six months already. This just isn’t fair.

Son, life isn’t fair. Ruthie isn’t sick because she did anything bad. She just got sick. Sometimes good things happen, sometimes bad things happen. All we can do is try to make a few more good things happen and a few less bad things happen. But a lot of things that happen to us, we can’t do anything about.

Dad, that’s really scary.

Yes, son, it really is.

That night I thought of a plan. I’d work very, very hard. Study all the time, no more hide-and-seek or silly stories told to little kids. I’d finish high school years early, go to college, then graduate school and medical school, become a medical researcher, find a cure for Hodgkin’s lymphoma, administer it to Ruthie, and save her. In one version of the fantasy, I flew in a helicopter from my university laboratory to Ruthie’s hospital. I was so pleased with my plan. I told Mom and said I’d tell Ruthie not to worry, that I’d save her. I expected Mom to be happy, but she looked very sad, told me I couldn’t tell Ruthie.

Why not? Don’t you want her to know she’ll be alright?

Mikey, I don’t want you to get her hopes up. A lie, but a gentle, sweet lie. No matter how hard you work, you might not be able to save Ruthie.

Logically I knew Mom was right. I’d gone to the library in Charleston, found an oncology book (I’d asked Mom the scientific name for the study of cancer), and found the Hodgkin’s survival statistics. They weren’t encouraging. But I wasn’t able to imagine a world without Ruthie. We had years of exploration still to do. And besides, how could Ruthie leave her sweet mother, Luverna Frame, the kindest, gentlest adult in my world? There had to be a way out of this, and I would find it.

But Ruthie died. Dad was at the hospital with her, holding her hand, when she died. When he came home, his expression told me all I needed to know. He told Mom, Linda, and Steve. They cried; I didn’t. Eventually Mom said that Ruthie had been terribly sick, would never be well again, so it was better that she wasn’t suffering anymore. Ruthie was suffering? Linda wailed. Then she and Steve began to run around and shriek. Eventually they calmed down to sobs. But I’d known that Ruthie was miserable. Waiting at the hospital hall outside her room while Dad checked that it was okay for me to come in, sometimes I heard her moan. She’d suffered, and now she didn’t. Was the peace of non-existence better than pain with little relief? Big puzzles for a twelve-year-old. Big puzzles still. . . .

Dad didn’t want us kids to go to the funeral. Mom and Dad went while we stayed with Mom’s parents, Burl and Lydia Arrowood. I found a bag of balloons in Gramp’s workshop. Gramp was a jeweler and repaired clocks and watches. Because he used a gas torch to melt some alloys, his workshop had a gas jet. I filled a balloon with gas, tied it off, walked into the front yard away from the trees and let the balloon fly. The symbolic content of this was melancholy: it represented all the experiments that Ruthie and I had planned to do, that now were lost forever. It represented the closing of a door.

So I closed myself off from the world. I could no longer help Ruthie, but maybe I could help other people. All I did was read and study science. Mom and Dad tried to get me to run around outside. They said Linda and Steve missed me, but I don’t think they did. All summer long they were outside, awakened by an early serenade of blue jays and catbirds, games of tag and hide-and-seek interrupted by dusk with its drifting fireflies. No, they didn’t need me.

And now I had a goal: I could no longer help Ruthie, but I could find cures for diseases and save other people. The determination of a serious twelve-year-old can be fierce, and I was fierce squared.

Later that year, I read a supplementary problem in my algebra text. For much of the weekend I tried all sorts of tricks. Eventually I found a solution, but it was clunky, mechanical, inelegant. It worked, but I knew it wasn’t the solution the author intended. After math class on Monday, I asked my teacher. She smiled, said she was happy I tried the problem, then wrote the simple, beautiful solution.

At that moment, my world folded in on itself, disappeared, and I knew what I thought was a different flavor of grief. The solution used only tricks I knew but applied them in a clever way that hadn’t occurred to me. At that moment, I began to suspect I was not bright enough to be a good scientist. Determination and hard work would get me into the tribe of scientists, but would life as a supporting character be enough? Choosing that path carried the real risk that from the end of life, where I am now, a backward look would reveal decades of steady work punctuated by very few moments of modest insight. To be sure, those moments have been amazing. The pleasure of understanding some bits of the architecture of ideas is ample reward. But I wanted to do so much more.

Has my life been so different from the lives of others? For some people, aptitude and interest align and a satisfying life unfolds, enviably free from regret or second-guessing. But many of us are haunted by thoughts of a path not taken. Some choices lead us along paths that we cannot reverse. Even if we change course now, what remains of our lives will not unfold as if we had made the other choice years earlier. What might have been is beyond our reach, and we grieve this loss.

For me, the path I choose—exploring some structures of math—has afforded new perspectives on grief. I believe grieving exhibits some similarities to doing math; we’ll find echoes of each in the other. Wrestling with mathematical questions has helped me to parse my own episodes of grief. That is my subject here.

In The Doubter’s Almanac, Ethan Canin writes:

Is the sorrow of death the same as the sorrow of knowing the pain in a child’s future? What about the melancholy of music? Is it the same as the melancholy of a summer dusk? . . . Both we call grief. . . .

But how to solve the grief I felt for my father in those last days? We think that our sorrow, like the planes we know in this world, has borders. But does it?²

Because geometry is, for me, the most beautiful part of math, and the part I know best, I’ll focus on geometry: the geometry of grief. This is as distinct from the grief of geometry—the longing to escape a late afternoon class where the teacher plodded through a proof of side-angle-side in two-column format on the chalkboard—as the lyric If it weren’t for bad luck, I’d have no luck at all is from Puccini’s Nessun dorma. In this book we’ll explore some ways in which grief informs geometry and geometry informs grief.

The architecture of this project was largely in place before I looked into what others had written. A notion that’s repeated often in this book is that an idea can’t be unseen. Taking in others’ ideas before thinking through my own experiences with grief might have limited how I understood those experiences. Only after I’d sketched out a rough draft did I read background studies on grief. Particularly useful were the evolutionary perspectives in psychologist John Archer’s The Nature of Grief, anthropologist Barbara King’s How Animals Grieve,